If [MATH]A = \begin{bmatrix}0 & 0 & \cdots & 0 & a_{1n}\\ 0 & 0 & \cdots & a_{2,n-1} & 0\\ & & \cdots & & \\ 0 & a_{n-1,2} & \cdots & 0 & 0\\ a_{n1} & 0 & \cdots & 0 & 0\end{bmatrix}[/MATH],
Show that [MATH]detA = (-1)^{k}a_{1n}a_{2,n-1} \cdots a_{n1}[/MATH], with [MATH]k = 0[/MATH] if [MATH]n = 4x[/MATH] or [MATH]n = 4x+1[/MATH] and [MATH]k = 1[/MATH] with [MATH]n = 4x + 2[/MATH] or [MATH]n = 4x + 3[/MATH]
The product [MATH]a_{1n}a_{2,n-1} \cdots a_{n1}[/MATH] is quite straight forward by the definition of the determinant and based on the properties of the matrix.
The [MATH](-1)^k[/MATH] part i don't understand because I think [MATH]k = (1 + n) + (1 + n - 1) + ( 1 + n - 2) + \cdots + (1 + 1)[/MATH] so the conditions of [MATH]k = 0[/MATH] and [MATH]k = 1[/MATH] don't make much sense to me...
The LaTeX matrix representation is funky, sorry about that...
By the way, I've been wondering if Intermediate/Advanced Algebra is the correct place to ask questions related to linear algebra?
Show that [MATH]detA = (-1)^{k}a_{1n}a_{2,n-1} \cdots a_{n1}[/MATH], with [MATH]k = 0[/MATH] if [MATH]n = 4x[/MATH] or [MATH]n = 4x+1[/MATH] and [MATH]k = 1[/MATH] with [MATH]n = 4x + 2[/MATH] or [MATH]n = 4x + 3[/MATH]
The product [MATH]a_{1n}a_{2,n-1} \cdots a_{n1}[/MATH] is quite straight forward by the definition of the determinant and based on the properties of the matrix.
The [MATH](-1)^k[/MATH] part i don't understand because I think [MATH]k = (1 + n) + (1 + n - 1) + ( 1 + n - 2) + \cdots + (1 + 1)[/MATH] so the conditions of [MATH]k = 0[/MATH] and [MATH]k = 1[/MATH] don't make much sense to me...
The LaTeX matrix representation is funky, sorry about that...
By the way, I've been wondering if Intermediate/Advanced Algebra is the correct place to ask questions related to linear algebra?